关于某类Briot-Bouquet微分方程若干结果的推广

本文研究了某类形式的Briot-Bouquet微分方程的亚纯解结构问题.利用Nevanlinna值分布理论的有关知识,获得了该类微分方程的所有可能形式,以及当方程的阶为偶数时,给出了每一个形式的方程的亚纯解结构,推广了某些特定的Briot-Bouquet微分方程亚纯解结构的一些结果.     

某些常微分方程的亚纯解表示与应用

本文运用Nevanlinna值分布理论研究了某些常微分方程亚纯解的存在性. 对于某些具有控制项的常系数常微分方程, 本文得到了亚纯解的表示, 并且给出了求相应偏微分方程精确解的一种方法.作为例子, 本文运用此方法得到了著名的KdV方程的所有亚纯行波精确解. 结果显示该方法比其他方法简单.     

The representation of meromorphic solutions for a class of odd order algebraic differential equations and its applications

In this paper, we employ the Nevanlinna's value distribution theory to investigate the existence of meromorphic solutions of algebraic differential equations. We obtain the representations of all meromorphic solutions for a class of odd order algebraic differential equations with the weak ?p,q?and dominant conditions. Moreover, we give the complex method to find all traveling wave exact solutions of corresponding partial differential equations. As an example, we obtain all meromorphic solutions of the Kuramoto–Sivashinsky equation by using our complex method. Our results show that the complex method provides a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics. Copyright © 2014 John Wiley & Sons, Ltd.     

On Meromorphic Solutions of Nonlinear Complex Differential Equations

Applying the Nevanlinna theory of meromorphic function,we investigate the non-admissible meromorphic solutions of nonlinear complex algebraic differential equation and gain a general result.Meanwhile,we prove that the meromorphic solutions of some types of the systems of nonlinear complex differential equations are non-admissible.Moreover,the form of the systems of equations with admissible solutions is discussed.     

Meromorphic solutions of some algebraic differential equations related Painlevé equation IV and its applications

In this paper, the complex method is used to derive meromorphic solutions to some algebraic differential equations related Painlevé equation IV, and then we illustrate our main result by some computer simulations. By the application of our result, we obtain meromorphic solutions of a nonlinear evolution equation. We can apply the idea of this study for other nonlinear evolution equations in mathematical physics.     

复微分方程组亚纯解的分量的Nevanlinna特征估计

本文讨论了一类复微分方程组的亚纯解的分量的Nevanlinna特征估计,利用亚纯函数Nevanlinna值分布理论,得到了一个有关方程组亚纯解的分量的Nevanlinna特征估计定理.该结果推广了高凌云等人的一些结果.     

All meromorphic solutions of an ordinary differential equation and its applications

Dedicated to Professor Yuzan He on the Occasion of his 80th Birthday In this paper, we employ the complex method to obtain all meromorphic solutions of an auxiliary ordinary differential equation at first and then find out all meromorphic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations. Our result shows that all rational and simply periodic exact solutions of the combined KdV–mKdV equation and variant Boussinesq equations are solitary wave solutions, the method is more simple than other methods, and there exist some rational solutions w_r,2(z) and simply periodic solutions w_s,2(z) that are not only new but also not degenerated successively by the elliptic function solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

二阶复域微分方程亚纯解的不动点与超级

研究了亚纯函数系数的二阶线性微分方程解的不动点及超级问题,得到了有关复域微分方程亚纯解的不动点性质,并且由于受到微分方程的制约,其性质与一般亚纯函数的不动点性质相比,显得十分有趣.     

Unicity of meromorphic solutions of partial differential equations

In this survey, results on the existence, growth, uniqueness, and value distribution of meromorphic (or entire) solutions of linear partial differential equations of the second order with polynomial coefficients that are similar or different from that of meromorphic solutions of linear ordinary differential equations have been obtained. We have characterized those entire solutions of a special partial differential equation that relate to Jacobian polynomials. We prove a uniqueness theorem of meromorphic functions of several complex variables sharing three values taking into account multiplicity such that one of the meromorphic functions satisfies a nonlinear partial differential equations of the first order with meromorphic coefficients, which extends the Brosch??s uniqueness theorem related to meromorphic solutions of nonlinear ordinary differential equations of the first order.     

亚纯函数系数高阶复域微分方程解的不动点

研究了一类亚纯函数系数的高阶线性微分方程的解的不动点问题,应用值分布的理论和方法,得到了复域微分方程亚纯解的不动点性质.     

On the meromorphic solutions of an equation of Hayman

The behavior of meromorphic solutions of differential equations has been the subject of much study. Research has concentrated on the value distribution of meromorphic solutions and their rates of growth. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi-parameter ordinary differential equation introduced by Hayman. This equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions—only those that are meromorphic. This is achieved by combining Wiman-Valiron theory and local series analysis. Hayman conjectured that all entire solutions of this equation are of finite order. All meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one.     

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.     

Existence of meromorphic solutions of some higher order linear differential equations

This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.     

二阶代数微分方程亚纯解的增长性估计

本文研究了代数微分方程亚纯解的增长级.运用正规族理论,给出了某类二阶代数微分方程亚纯解的增长级的一个估计,该估计依赖于方程的有理函数系数.推广了2001年廖良文与杨重骏的一个结果.     

复非线性代数微分方程解的增长级(英文)

本文研究了一类非线性微分方程的解的增长级的问题.利用亚纯函数的Nevanlinna值分布理论和Wiman-Valiron整函数理论的方法,获得了比以往文献更为精确,更为一般的结论,推广了Gol’dberg,Barsegian,Hayman以及Korhonen等作者的一些结果.     

On the growth of meromorphic solutions of the Schwarzian differential equations

We study the properties of meromorphic solutions of the Schwarzian differential equations in the complex plane by using some techniques from the study of the class Wp. We find some upper bounds of the order of meromorphic solutions for some types of the Schwarzian differential equations. We also show that there are no wandering domains nor Baker domains for meromorphic solutions of certain Schwarzian differential equations.     

Über den Satz von Poincaré in der Störungstheorie Briot-Bouquetscher Differentialgleichungen

The well known theorem of H. Poincaré concerning the analytic continuation and representation along a curve of solutions of differential equations depending on a parameter is generalized to solutions of Differential Equations of Briot-Bouquet type depending on a parameter.

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Herrn Professor Dr. N. Hofreiter zum 70. Geburtstag     

Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations

Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations.     

On global meromorphic solutions of second-order linear differential equations with meromorphic coefficients

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.     

二阶复微分方程的亚纯元素解及代数元素解

研究了一类二阶亚纯系数复微分方程的亚纯元素解及代数元素解的存在性问题,得到了几个有关解的存在性定理.